Problem: Assume that the earth is a sphere of radius $5280$ miles, find the length of the sides, the measure of the angles and the area of the spherical triangle with vertices $A(70°N,10°E)$,$B(10°S,100°E)$ and $C(50°S,80°W)$. The earth's radius will be used as the unit of length. The spherical coordinates $(r,v,u)$ of the three vertices are $(1,10,20)$, $(1,100,100)$ and $(1,−80,140).$
Their Cartesian coordinates are:
$$(\sin(20°)\cos(10°),\sin(20°)\sin(10°),\cos(20°))=(0.3368,0.0594,0.9397)$$
$$(\sin(100°)\cos(100°),\sin(100°)\sin(100°),\cos(100°))=(−0.1710,0.9698,−.1736)$$
$$(\sin(140°)\cos(−80°),\sin(140°)\sin(−80°),\cos(140°))=(0.1116,−0.6330,−.7660)$$
The cosines of the angles between radii $OA,$ $OB$ and $OC$ are equal to the dot product of the corresponding position vectors. Thus:
$$∠AOB=\cos^{−1}(-0.1631)=1.7347 \ \mathrm{radians}$$
$$∠BOC=\cos^{−1}(-0.5000)=2.0944 \ \mathrm{radians}$$
$$∠COA=\cos^{−1}(-0.7198)=2.3743 \ \mathrm{radians}$$
Since the length of the arc of a circle is the product of its radius by the radian measure of its central angle, it follows that the length of the sides of the spherical triangle $ABC$ are:
$9,159$ miles, $11,058$ miles, and $12,536$ miles.
The angle of this triangle are derived by means of the appropriate Law of cosines. Thus:
!This is where I begin to have problem understanding the question!!
$$\cos A = \frac{\cos(2.0944) – \cos(1.7346) \cos(2.3743)}{\sin(1.7346) \sin(2.3743)}
= -0.9013$$
So $A=154°=2.6941$ radians
Similarly,
$B=160°=2.7874$ radians
$C=150°=2.6261$ radians
So, my question is how did they calculate $B$ and $C$ to be $160°$ and $150°$?
The book says:
$$\cos A = \frac{\cos(a)-\cos(b) \cos(c)}{\sin(b) \sin(c)}.$$
Therefore, I know how they calculated A. So i'm guessing they did $B$ and $C$ the same way but I dont get the same answer.
SO
$$a=2.0944$$
$$b=1.7346$$
$$c=2.3743$$
So if I wanted to calculate $B$:
$$\cos B = \frac{\cos(b)-\cos(a) \cos(c)}{\sin(a) \sin(c)}.$$
$$\cos B = \frac{\cos(1.7346)-\cos(2.0944) \cos(2.3743)}{\sin(2.0944)\sin(2.3743)}.$$
and I get $B=45°$ not $160°$
I'm not sure where I went wrong.
Best Answer
You miscalculated and you confused the values for Cos B for C. David said it already but make sure your calculator is in radian mode. You have the calculation and values right, but its Cos C. I checked in radians mode and it seem to work out.